Differential calculus for Dirichlet forms: The measure-valued gradient preserved by image

In order to develop a differential calculus for error propagation of Bouleau [Error Calculus for Finance and Physics, the Language of Dirichlet forms, De Gruyter, Berlin, 2003], we study local Dirichlet forms on probability spaces with carré du champ —i.e. error structures—and we are looking for an object related to which is linear and with a good behaviour by images. For this we introduce a new notion called the measure-valued gradient which is a randomized square root of . The exposition begins with inspecting some natural notions candidate to solve the problem before proposing the measure-valued gradient and proving its satisfactory properties. © 2005 Elsevier Inc. All rights reserved.